Differentiation practice · Mixed difficulty

Option D (None of these) is correct because the nth derivative of the product of cosine functions involves a sum of cosine functions with arguments adjusted by nπ/2, and the given options do not match the correct form.

Option A ($a=-3 / 4, b=-1 / 8$) is correct because these values satisfy the conditions for the function to have extrema at x=1 and x=3.

Option C ($1 + {\{\phi (x)\}^5}$) is correct because it is derived from the inverse function differentiation rule.

The correct answer is C ($4 x(1-x^{2}) \sin (1-x^{2})^{2}$) because it is the result of applying the chain rule multiple times to differentiate the given function.

Option C ($\left| y''(0) \right| = 2$) is correct because differentiating the given equation and evaluating at x=0 yields |y''(0)| = 2.

Option B is correct because it correctly uses the given conditions to determine the sign of the quadratic function g(x).

Option B ($\frac{1}{1+x^2}$) is correct because after substituting $x = \tan \theta$ and simplifying the expression, the derivative is found to be $\frac{1}{1+x^2}$.

The correct answer is A ($-\frac{1}{4}$) because after simplifying the expression inside the inverse tangent, the derivative is a constant.

Option B ($\frac{1}{2}$) is correct because it is the value of the derivative $f'(2)$ after solving for $f(x)$ and differentiating.

Option B ($2^n$) is correct because the given expression is the binomial expansion of $(1+1)^n$, which equals $2^n$.

Option A ($\frac{2}{3}$) is correct because by substituting $x = \sin \theta$, the expression simplifies, and the derivative becomes a simple ratio.

Option C (8) is correct because it follows from the chain rule applied to composite functions.

Option A ($2 x^{\left(\log _a x-1\right)} \cdot \log _e x$) is correct because by applying logarithmic differentiation, we can find the derivative of the given function.

Option C (Non-existent) is correct because the left-hand derivative and right-hand derivative of the function at $x = \frac{\pi}{2}$ are not equal.

Option C ($\frac{y}{x}$) is correct because it is the derivative dy/dx obtained after simplifying the given equation using trigonometric identities and differentiating implicitly.

Option D (Does not exist) is correct because the left-hand derivative and right-hand derivative at $x = 0$ are not equal.

Option C ($\frac{{1 - x}}{x}$) is correct because it is the derivative of y with respect to x after taking the logarithm and differentiating.

Option A ($\frac{\log _2 \mathrm{e}}{\mathrm{x} \ln \mathrm{x}}$) is correct because it is the result of applying the chain rule and change of base formula to differentiate the given logarithmic function.

Option D (1) is correct because after simplifying the expression inside the arctangent, the derivative is 1.

Option C ($1 + {\left[ {\phi \left( x \right)} \right]^5}$) is correct because it is the derivative of the inverse function, calculated using the given derivative of the original function.

Option C is correct because the derivative of arcsin(3x-4x³) is piecewise, being 3/sqrt(1-x²) for |x|<1/2 and -3/sqrt(1-x²) for 1/2<|x|<1 due to the properties of arcsin.

Option C (100) is correct because f'(1) = 100 and f'(0) = 1, so f'(1) = 100f'(0).

Option D (8) is correct because by using the given functional equation and derivative information, we can find f'(4).

Option B ($\frac{{1}}{{2}}$) is correct because it is the value of $f^{\prime}(2)$ obtained by solving the system of equations after differentiating the given functional equation.

Option D is correct because using the given condition $f\left( {\frac{x}{y}} \right) = \frac{{f(x)}}{{f(y)}}$ and $f^{\prime}(1)=2$, we find $f^{\prime}(x) = \frac{2 f(x)}{x}$.

The correct answer is C ($\frac{{{y^2}\log y}}{{x\left( {1 - y\log x\log y} \right)}}$) because it is the result of logarithmic differentiation and solving for $\frac{dy}{dx}$.

Option A (0) is correct because after differentiating the given equation and simplifying, the expression evaluates to 0.

The correct answer is B (1) because applying L'Hopital's rule simplifies the limit to 1.

Option D ($4-65\left(\frac{1}{2}\right)^{62}$) is correct because it is the value of the derivative at x=1/2 after simplification.

Option D ($\frac{3}{{2\left( {1 + x} \right)\sqrt x }}$) is correct because after substituting and simplifying, the derivative equals $\frac{3}{{2\left( {1 + x} \right)\sqrt x }}$.